ON THE COLOURS OF THICK PLATES. 161 surface, R the retardation of the stream which was scattered at P on emergence, relatively to that which was scattered at the same point on entrance. Let i,, L2, L3 be the images of L after refraction, reflexion, and second refraction, respectively; M19 Mz, MSj the images of M; and let a, 6, c, or a, &', c', with the suffixes 1, 2, 3, denote the co-ordinates of i1? Z2, Z3, or J^, If2, MB. In approximating to the value of R, let the squares of the small quantities, a, 6, a?, y, &c. be retained, so that the terms neglected are of the fourth order, since all the terms are of even orders, as will be immediately seen when the approximation is commenced. 2. The rays diverging from L may after refraction be supposed to diverge from Llf notwithstanding the spherical aberration of direct pencils, and the astigmatism of oblique pencils. For, first, let L be in the axis. The supposition that the rays diverge from Ll is equivalent to supposing that the front of a wave is a sphere having L^ for centre, whereas it is really a surface of revolution such that LI is the centre of curvature of a section made by a plane through the axis. This plane cuts the sphere above mentioned in a circle, which, being a circle of curvature, cannot have with the curve a contact lower than one of the second order. But the contact is actually of the third order, since the curve and circle touch without cutting. Hence the error produced in tVie calculation of R by supposing the front of a wave to be a sphere, instead of that surface which it actually is, is only a small quantity of the fourth order, and quantities of this order are supposed to be neglected. Next, consider an oblique pencil. Let L' and L" be two points in the axis of the pencil which are the centres of curvature of its principal sections. If the distance of L' and L" from each other, arid from Llf were not small, the front of the wave would have a contact of the first order with a sphere described round Llt with such a radius as to pass through the point where the front is cut by the axis of the pencil; and in that case the error committed by taking the sphere for the actual front would be of the second order. But L and L" are situated at distances from Ll which are small quantities of the second order, whence it will readily be seen that the actual error is only of the fourth order. 3. Let the expression (L to L3) denote the retardation of a wave proceeding from L to L3, or rather, in case £3 be a virtual s. in. I!